A (hierarchical) ranking is always based on 'things'. Such things can be of different kinds, and are for example called a 'particle', an 'object', a 'token', an 'individual' or a 'group'. In fact, there is a lot of confusion about such concepts. In order to reduce confusion and ambiguity, we suggest to construct concepts 'from the ground up'.

Defining the entities that play a role in hierarchies (particles, objects, tokens and individuals)

Often a hierarchy is constructed from entities that are identified using the form in which they appear to us (an a posteriory identification). For example an apple can be described as a separate entity, for example: "a green or reddish fruit that has a rounded shape, etc.". A complementary approach is to define the apple from the bottom up, by looking at its origin (an a priory identification). Now the apple is viewed as the kind of fruit that has grown on a tree, which we cal an apple tree. Of course, entities have been defined both by a priory and a posteriory methods. As will be shown, this duality causes confusion if one wants to rank entities in a hierarchical way. As many entities of different kinds are used interchangingly, such as 'particles', 'objects', 'tokens' and 'individuals', this leads to confusion about rankings. Here the meanings of these concepts will be discussed first, after which it will be shown that one can construct, from the bottom up, a ranking in which, in a recurrent way, the organisation of entities at level X forms the basis for the organisation of the entities at level X+1.

Token.

In philosophy, the concept of a 'token' is used to indicate a single physical entirety that shows an outer limit and exists at a certain moment at a certain place. A token can be counted as a single entity, and can normally be displaced as a single entity. If a selected token has parts, these must be physically connected or linked, such that they are part of the selected token. A group of objects which are not physically (materially) connected is never a token. Examples of tokens are: an apple, a bicycle, and an atom (the electron shell is viewed now as a probability boundary defining the outer limit of the atom). A colony of bees, or a herd of deer, is not a token, because every bee or deer is a separate token. As will be shown in this website, there exists a special way of defining a certain hierarchy of tokens from the ground up. Working from the ground up solves the problem that tokens are normally defined a posteriory: we see an 'object' and try to define it as a token. While working from the ground up, one can define certain tokens in an a priory way (specific interactions between certain existing tokens lead to certain new kinds of tokens).

Particle.

The concept of a 'particle', is associated with a small lump of matter (a dust particle, a piece of brick) and with the historical discoveries of increasingly small physical particles. First the atoms ('a-tomos' indicates "that what cannot be cut") were considered the smallest. Then atom nuclei were discovered. And later discoveries revealed that nuclei harbour protons and neutrons, which in turn harbour quarks and gluons. The so called 'Standard Model' offers an overview of the 'particles' which already for a long time, and for various reasons, are viewed as fundamental.

Object.

We distinguish between two types of objects. First, every subset of physically connected parts of the universe (token/particle) can be called a physical object. This includes for example colonial organisms that grow together, such as the Portuguese ship of war. Another perspective is possible, if we set our mind to an 'object oriented' attitude. Now we can additionally view a group of tokens as a mental object (e.g. the Andromeda nebula, or the Milky Way galaxy). Similarly we can view a specific herd of deer, or a specific human society as a mental object. Yet, a galaxy, a herd, or a human society cannot be viewed as a token or as a particle.

Individual

The concept of the 'individual' means 'something that cannot be divided'. Whether an entity ‘cannot be divided’ may vary with a choice for specific criteria, which makes 'individuality' an ambiguous concept. Generally, 'individuality' refers to some kind of holistic property which can be lost after division. For example a piece of brick, as an individual, can be broken into parts, which again are individual pieces of brick. But if an atom is divided into a nucleus and an electron shell, one does not obtain two individuals of the type atom. Moreover, people have a natural tendency to view a herd of dear, or a human society as a mental objects, and therefore sometimes as a mental individual. However, a group can be devided in two groups, which both can be viewed as mental individuals. For this reason a group is 'divisible', instead of being 'indivisible'. For this reason it is preferred here to strictly view a group as a group, and not as an individual.

Operator and organism.

The 'operators' are introduced below, and can be used to define (from the ground up) the concept of the organism, as is discussed here (definition of organism and life).

Figure 1: Comparison of concepts that are in use for indicating different forms of 'entities'. A + sign indicates that the concept can be applied without major problems. A minus indicates that the concept does not apply. The +/- signs indicate ambiguities in interpretation.

Solving ambiguities by constructing concepts from the ground up

If we want to reduce ambiguities resulting from concepts such as particle, object, token and individual, it may help to develop a new approach to objects and their hierarchy. For this new approach, we want to define every level, and the related type of entities at that level, from the bottom up. Moreover, we consider fixed hierarchical levels a necessity, because only fixed levels can associate entities with fixed positions in a hierarchy.

To create fixed levels, we use the organisation of lower level entities as a basis for the organisation of higher level entities. When focusing on the kind of organisation of lower level entities, we do not use the structure of lower level entities in a physical sense, but in a topological sense. It is proposed that transitions towards a next type of object can be defined by means of topology, by using a combination of first possible, next, structural and functional topology for defining any next level object.

Structural closure

The following example can illustrate the use of topology for the identification of levels of organisation. Suppose we start a hierarchy with a simple topological shape: a balloon. This is a three dimensionally closed surface with no hole in it (while a doughnut or the number eight are closed surfaces with holes). Based on its closed shape, we call the balloon a ‘closed object’. A balloon, as a closed object, can now be used to illustrate the construction of a hierarchy. As is shown in Figure 1 the strucutral combination of balloons can result in three different shapes. The first (B1) is that of a balloon which is attached to another balloon (they share an 'in between' surface). The second (B2) is that of a balloon which partially engulfs the other balloon. The third (B3) is that of a balloon which has entirely engulfed another balloon. Topology now assists in reducing the number of options, because the partially engulfed situation of (B2) is identical –in a topological sense- to the attached situation of (B1). A topological analysis thus helps in reducing ambiguity and in creating clear types (attached versus entirely engulfed). The example shows that the structure of a balloon can serve as the basis for precisely two next structures in our hierarchy, namely attached balloons and entirely engulfed balloons. Both new situations are exactly one 'closure step' away from the initial balloon shape. And both situations are the immediate next possibilities for a new type of closure situation. Therefore, both new topologies comply with what we refer to as ‘first possible next new kind of closure’, in short; first possible next closure, or FPNC.

Figure 1: A thought experiment that shows how toplogy can be used for analysing the possibilities of using a 'balloon' as the basis for new structures that are based on two 'balloons'

In principle one may also suggest that a group of separate balloons also represents a topological possibility. While this is true, such a group of not-connected balloons would not count as a structurally closed topology. In this way, structural closure allows for a clear distinction between 'objects with structural closure' and groups of such object, which as a group fail to show structural closure.

Functional closure

However important shapes are, they are not the only patterns causing structure in nature. Also processes may play a role in the formation and typing of entities. Therefore, we apply topology also to processes. For example a person could –as a process- lay his left and right hand on the shoulders of two other persons. If these persons do the same, in an iterative way, this results in the topology of a binary branching tree. But a process can also be self-referential, or recurrent, which implies a functional closere. For example if a person whispers a message in an ear of a next person, this may result in a process cycle in which the last person finally whispers the message again in the ear of the first person, hereby closing the process (Hoffstadter would say that a ‘strange loop’ was created).

Defining a new type of object with a closed topology, by combining structural and functional closure

Is it possible to identify a minimum set of necessary and sufficient conditions for defining a next level object if we use topological criteria? According to the current viewpoint, a next level objects of similar type (namely of the type: 'double closed') can be defined from the ground up. For this purpose one can use an integration of a structural and a functional closure. The structural closure causes the next type of entity to possess physical/material/constructional unity. And the functional closure assures that the new type of entity possesses a new functionality which was not possible at the preceding level. It is hypothesized that the combination of these two closures is both minimal, necessary and sufficient for defining the transition to a first-possible, new closure situation. In turn, such a first-possible new closure defines the 'next level object'. The following two examples introduce this way of thinking. More detailed information is offered in the next page of this website, where specific attention is paid to the different kinds and combinations of closures.

The first example we discuss starts at the level of the atom. The question is now, what next 'object' can be imagined at the 'level' immediately above the atom? How can one define a new object at this 'level to come', and implicitly define the criteria for entities at this level? If we use functional and structural closure, we can start with observing that every single atom has its electron shell as its proper structural closure. We now need to identity a next structural and functional closure that leads to the integration of two or more atoms. A property which typically emerged with the existence of atoms, is that the electrons in the electron orbits have the tendency to form pairs. If an orbit is inhabitated by a single electron, this forms a relatively unstable state. As soon as another atom with a single electron comes within reach, these electrons will 'condense' to form a pair. But the electron pair cannot be part of both the atoms at the same time. Charge differences now cause an exchange of the electron pair, which causes attraction between the atoms, called covalent bonding. One can now view the shared orbit for the paired electrons as a new structural closure, and one can view the exchange of the electron pair as a new functional closure. Both the paired orbit and the exchange between atoms are properties that are not found in single atoms. Both closures necessarily lead to the formation of a molecule (as a multi-atom structure), and are also sufficient to define a molecule. One needs these closures to distinguish an atom from a molecule in a stringent way.

The second example we discuss starts at the level of the molecule. Any molecule has its multi-atomary three-dimensional structure as its new property. Based on this three dimensional structure it is possible to form both a next structural and a next functional closure. The next structural closure must involve a closed topology based on molecule-molecule interactions. The simplest topology of this kind is a globule that is formed of a sheet of mutually adhering/attracting molecules. Such globlues are readily formed as a condensation product of e.g. fatty acids in water. The simplest functional closure could involve two molecules A and B of which A catalyses a reaction which produces B, while B catalyses a reaction which produces A. In this way A and B produce 'setwise autocatalysis'. In fact, the two-closure situation is an oversimplification, because the molecules A and B must also catalyse reactions which lead to more molecules for the cell membrane (e.g. Blundell). Now the process cycle of catalytic process is closed for the production of A and B. The process in which A catalyses the production of A (a 'uni-moleculary autocatalysis') is excluded from the present reasoning, because this process already existed at the level of single molecules (it is not of a 'new' type). It is the combined process of A and B which is required for a new type of closed process. The interaction between the globular membrane (structural closure) and the set of catalytic molecules (functional closure) causes a system which can autonomously produce its overall structure. The combination of structural and functional closure is necessary and sufficient to define the new system. Such a system is known as a (bacterial/prokaryote) cell.

Constructing a non-transitive hierarchy from the bottom up

If we now assume that fundamental particles are the smallest ‘balloons’ that exist in nature, we can start constructing closed functional and/or structural topologies from these fundamental ‘balloons’. How this works has been explained step by step in pages 36-55 of “The operator hierarchy. A chain of closures linking matter, life and artificial intelligence” (Alterra Scientific Contribution 34, 2010). A detailed explanation can furthermore be found online at www.hypercycle.nl. These sources use closure to explain every single step in the ranking from fundamental particles (as in the standard model) to hadrons, atoms and molecules, bacterial cells, endosymbiont cells, multicellulars and multicellulars with neural networks.

Operator types and the operator hierarchy

In order to refer to all the entities in the ranking of closures and closed entities described above, from fundamental particles to organisms with a neural network, it is practical to introduce a new concept: the operator. This name has been chosen to indicate in a general way that every operator can ‘operate’ (move, be displaced, act, etc.) in its environment while its closures persist. Accordingly, one can always use the closures (at the respective levels) as a hallmark of whether or not an entity is of the type ‘operator’. In relation to this, the ranking of types of closures, is called the operator hierarchy. For every closure type, there exist real world entities, the ‘operators’ that form the instantiations of the type.